Two sides of a triangle are 16.2 cm and 29.6 cm, with the included angle 106◦. What is angle opposite the side 16.2 cm?

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Answer 1 · 2016-12-02T17:37:01

The angle $A ~~ 24.6^@$

Let side $a = 16.2" cm"$
Let side $b = 29.6" cm"$
Let side $c =$ the unknown side.

Let $angle A =$ the unknown angle opposite side a.
Let $angle B =$ the unknown angle opposite side b.
Let $angle C = 106^@$

Use the Law of Cosines to find the length of side c:

$c = sqrt(a^2 + b^2 - 2(a)(c)cos(C))$

$c = sqrt((16.2" cm")^2 + (29.6" cm")^2 - 2(16.2" cm")(29.6" cm")cos(106^@))$

$c ~~ 37.46" cm"$

Use the Law of Sines to find $angle A$

$sin(A)/a = sin(C)/c$

$sin(A) = sin(C)a/c$

$A = sin^-1(sin(C)a/c)$

$A = sin^-1(sin(106^@)(16.2" cm")/(37.46" cm"))$

$A ~~ 24.6^@$